It was a clever idea: to put out a guide to the Moon in the same format as one of Haynes’ famous car-maintenance manuals. And the execution matched the idea. This is a detailed and interesting history of selenological speculation and lunar exploration, all the way from the ancient Greeks to the Apollo missions and beyond.

Except that there hasn’t been much beyond the Apollo missions. As the book’s final page notes:

On 31 December 1999 National Public Radio in the United States asked Sir Arthur C. Clarke, renowned for forecasting many of the developments of the 20th century, whether anything had happened in the preceding 100 years that he never could have anticipated. “Yes, absolutely,” he replied without a moment’s hesitation. “The one thing that I never would have expected is that after centuries of wonder and imagination and aspiration, we would have gone to the Moon… and then stopped.” (“Postscript”, pg. 172)

And we’ve been stopped for some time. Neil Armstrong died in 2012, forty-three years after that “small step for a man” and “giant leap for mankind” in 1969. But David M. Harland ends on an optimistic note: he thinks that “The Moon is humanity’s future.” It will be our gateway to the rest of the solar system and perhaps even the stars.

But it will be more than just a gateway. There is still a lot we don’t understand about our nearest celestial neighbour and big surprises may still be in store. One thing we do now understand is that the scarred and pitted lunar surface got that way from the outside, not the inside. That is, the moon was bombarded with meteors, not convulsed by volcanoes. But that understanding, so obvious in hindsight, took a long time to reach and it was actually geologists, not astronomers, who promoted and proved it (ch. 5, “The origin of lunar craters”). It was the last big question to be settled before the age of lunar exploration began.

Previously scientists had looked at the Moon with their feet firmly on the ground; at the end of the 1950s, they began to send probes and robotic explorers. Harland takes a detailed look at what these machines looked like, how they worked and where they landed or flew. Then came the giant leap: the Apollo missions. They were an astonishing achievement: a 21st-century feat carried out with technology from the 1960s, as Harland puts it. Yet in one way they depended on technology much earlier than the 1960s: pen and paper. The missions relied on the equations set out in Newton’s Principia Mathematica (1687). Newton had wanted to explain, inter multa alia, why the Moon moved as it did.

By doing that, he also explained where a spacecraft would need to be aimed if it wanted to leave the Earth and go into orbit around the Moon. His was a great intellectual achievement just as the Apollo missions were a great technological achievement, but he famously said that he was “standing on the shoulders of giants”. Harland begins the book with those giants: the earlier scientists and mathematicians who looked up in wonder at the Moon and tried to understand its mysteries. Apollonius, Hipparchus and Ptolemy were giants in the classical world; Galileo, Brahe and Kepler were giants in the Renaissance. Then came Newton and the men behind the Apollo missions.

Are there more giants to come? The Moon may be colonized by private enterprise, not by a government, so the next big names in lunar history may be those of businessmen, not scientists, engineers and astronauts. But China, India and Japan have all begun sending probes to the Moon, so their citizens may follow. Unless some huge disaster gets in the way, it’s surely only a matter of time before more human beings step onto the lunar surface. Even with today’s technology it will be a great achievement and more reason to marvel at the Apollo missions. And the Apollo photographs still look good today.

There are lots of those photographs here, with detailed discussion of the men and machines that allowed them to be taken. The Moon is a fascinating place and this is an excellent guide to what we’ve learned and why we need to learn more.

The Invention of Science: A New History of the Scientific Revolution, David Wootton (Allen Lane 2015)

I picked up this book expecting to start reading, then get bored, start skimming for interesting bits, and sooner or later give up. I didn’t. I read steadily from beginning to end, feeling educated, enlightened and even enthralled. This is intellectual history at nearly its best, as David Wootton sets out to prove what is, for some, a controversial thesis: that “Modern science was invented between 1572, when Tycho Brahe saw a new star, and 1704, when Newton published his Opticks” (introduction, pg. 1).

He does this in a clever and compelling way: by looking at the language used in science across Europe. If there was indeed a scientific revolution and science was indeed a new phenomenon, we should expect to see this reflected in language. Were old words given new meanings? Did new words and phrases appear for previously inexpressible concepts? They were and they did. “Scientist” itself is a new word, replacing earlier and less suitable words like “naturalist”, “physiologist”, “physician” and “virtuoso”. The word “science” is an example of an old word given a new meaning. In Latin, scientia meant “knowledge” or “field of learning”, from the verb scire, “to know”.

But it didn’t mean a systematic collective attempt to investigate and understand natural phenomena using experiments, hypotheses and sense-enhancing, evidence-gathering instruments. Science in that sense was something new, Wootton claims. He assembles a formidable array of texts and references to back his thesis, which is part of why this book is so enjoyable to read. As Wootton points out, the “Scientific Revolution has become almost invisible simply because it has been so astonishingly successful.” Quotations like this, from the English writer Joseph Glanvill, make it visible again:

And I doubt not but posterity will find many things, that are now but Rumors, verified into practical Realities. It may be some Ages hence, a voyage to the Southern unknown Tracts, yea possibly the Moon, will not be more strange then one to America. To them, that come after us, it may be as ordinary to buy a pair of wings to fly into remotest Regions; as now a pair of Boots to ride a Journey. And to conferr at the distance of the Indies by Sympathetick conveyances, may be as usual to future times, as to us in a litterary correspondence. (The Vanity of Dogmatizing, 1661)

Glanvill’s prescience is remarkable and he’s clearly writing in an age of pre-science or proto-science. He wasn’t just a powerful thinker, but a powerful writer too. So was Galileo and Wootton, who has written a biography of the great Italian, conveys his genius very clearly in The Invention of Science. You can feel some of the exhilaration of the intellectual adventure Galileo and other early scientists embarked on. They were like buccaneers sailing out from Aristotle’s Mediterranean into the huge Atlantic, with a new world before them.

Wootton also emphasizes the importance of Galileo’s original speciality:

The Scientific Revolution was, first and foremost, a revolt by the mathematicians against the authority of the philosophers. The philosophers controlled the university curriculum (as a university teacher, Galileo never taught anything but Ptolemaic astronomy), but the mathematicians had the patronage of princes and merchants, of soldiers and sailors. They won that patronage because they offered new applications of mathematics to the world. (Part 2, “Seeing is Believing”, ch. 5, “The Mathematization of the World”, pg. 209)

But there’s something unexpected in this part of the book: he describes “double-entry bookkeeping” as part of that mathematical revolt: “the process of abstraction it teaches is an essential precondition for the new science” (pg. 164).

He also has very interesting things to say about the influence of legal tradition on the development of science:

Just as facts moved out of the courtroom and into the laboratory, so evidence made the same move at around the same time; and, as part of the same process of constructing a new type of knowledge, morality moved from theology into the sciences. When it comes to evidence, the new science was not inventing new concepts, but re-cycling existing ones. (Part 3, “Making Knowledge”, ch. 11, “Evidence and Judgment”, pg. 412)

Science was something new, but it wasn’t an ideology ex nihilo. That isn’t possible for mere mortals and Wootton is very good at explaining what was adapted, what was overturned and what was lost. Chapter 13 is, appropriately enough, devoted to “The Disenchantment of the World”; the next chapter describes how “Knowledge is Power”. That’s in Part 3, “Birth of the Modern”, and Wootton wants this to be a modern book, rather than a post-modern one. He believes in objective reality and that science makes genuine discoveries about that reality.

But he fails to take account of some modern scientific discoveries. The Invention of Science is a work of history, sociology, philology, and philosophy. It doesn’t discuss human biology or the possibility that one of the essential preconditions of science was genetic. Modern science arose in a particular place, north-western Europe, at a particular time. Why? The Invention of Science doesn’t, in the deepest sense, address that question. It doesn’t talk about intelligence and psychology or the genetics that underlie them. It’s a work of history, not of bio-history or historical genetics.

In 2016, that isn’t a great failing. History of science hasn’t yet been revolutionized by science. But I would like to see the thesis of this book re-visited in the light of books like Gregory Clark’s A Farewell to Alms (2007), which argues that the Industrial Revolution in England had to be preceded by a eugenic revolution in which the intelligent and prudent outbred the stupid and feckless. The Invention of Science makes it clear that Galileo was both a genius and an intellectual adventurer. But why were there so many others like him in north-western Europe?

I hope that historians of science will soon be addressing that question using genetics and evolutionary theory. David Wootton can’t be criticized for not doing so here, because bio-history is very new and still controversial. And he may believe, like many of the post-modernists whom he criticizes, in the psychic unity of mankind. The Invention of Science has other and less excusable flaws, however. One of them is obvious even before you open its pages. Like Dame Edna Everage’s bridesmaid Madge Allsop, it is dressed in beige. The hardback I read does not have an inviting front cover and Wootton could surely have found something equally relevant, but more interesting and colourful.

After opening the book, you may find another flaw. Wootton’s prose is not painful, but it isn’t as graceful or pleasant to read as it could have been. This is both a pity and a puzzle, because he is very well-read in more languages than one: “We take facts so much for granted that it comes as a shock to learn that they are a modern invention. There is no word in classical Greek or Latin for a fact, and no way of translating the sentences above from the OED [Oxford English Dictionary] into those languages.” (Part 3, “Facts”, pg. 254)

He certainly knows what good prose looks like, because he quotes a lot of it. But his own lacks the kind of vigour and wit you can see in the words of, say, Walter Charleton:

[I]t hath been affirmed by many of the Ancients, and questioned by very few of the Moderns, that a Drum bottomed with a Woolfs skin, and headed with a Sheeps, will yeeld scarce any sound at all; nay more, that a Wolfs skin will in short time prey upon and consume a Sheeps skin, if they be layed neer together. And against this we need no other Defense than a downright appeal to Experience, whether both those Traditions deserve not to be listed among Popular Errors; and as well the Promoters, as Authors of them to be exiled the society of Philosophers: these as Traitors to truth by the plotting of manifest falsehoods; those as Ideots, for beleiving and admiring such fopperies, as smell of nothing but the Fable; and lye open to the contradiction of an easy and cheap Experiment. (Physiologia Epicuro-Gassendo-Charltoniana, 1654)

The Invention of Science is also too long: its message often rambles home rather than rams. If Wootton suffers from cacoethes scribendi, an insatiable itch to write, then I feel an itch to edit what he wrote. It’s good to pick up a solid book on a solid subject; it would be even better if everything in the book deserved to be there.

But if the book weren’t so good in some ways, I wouldn’t be complaining that it was less than good in others. In fact, I wouldn’t have finished it at all and I wouldn’t be heartily recommending it to anyone interested in science, history or linguistics. But I did and I am. The Invention of Science is an important book and an enjoyable read. I learned a lot from it and look forward to reading it again.

Sextant: A Voyage Guided by the Stars and the Men Who Mapped the World’s Oceans, David Barrie (William Collins 2014)

When a triumphant emperor rode through Rome, he’s said to have had a slave at his shoulder whispering: “Remember, Caesar, thou art mortal.” This book has a related message for its readers: “Remember, you’re comfortable.” The world has become much smaller and much safer since the days when a sextant was an essential part of every ship’s equipment.

Or has seemed to become smaller and safer, anyway. David Barrie reminded himself of the underlying reality by sailing across the Atlantic in 1973 with two companions in a 35-ft sloop called Saecwen (Anglo-Saxon for “Sea-Queen”). The voyage was powered by the wind and guided by the heavens in the old-fashioned way:

Of course I was intellectually aware of the size of the ocean when we set out from Halifax [on the coast of Nova Scotia], but spending twenty-four days crossing it under sail gave its dimensions a very different and truly sublime reality. The long night watches looking up at the stars in the black immensity of space were a lesson in humility and the experience of a gale in mid-Atlantic left me wondering what it must be like to encounter a real storm. People often talk idiotically about “conquering mountains” or “defying the sea”, but there is no real contest. I was left with an overwhelming sense of nature’s vast scale and complete indifference, and this had a strangely calming effect. We come and we go, the earth too was born and will eventually die, but the universe in all its chilly splendour abides. (ch. 18, “Two Landfalls”, pp. 289-90)

That’s at the end of the book. Descriptions of Barrie’s voyage in the 1970s open almost every previous chapter and set the context first for a history of celestial navigation and then for the stories of the men who used it. Their expertise with sextants and other instruments won them fame, but not always fortune. Nor a quiet and dignified death. Captain Cook charted the Pacific, then was hacked to death on Hawaii in 1779. Joshua Slocum made the first solo circumnavigation of the world in 1895-6, then “disappeared at sea after setting sail from Martha’s Vineyard on a single-handed voyage to the Amazon in November 1908” (ch. 15, “Slocum Circles the World”, pg. 255).

George Bass, after whom the strait separating Tasmania from Australia is named, disappeared too, perhaps at sea, perhaps into the slave-mines of a Spanish colony in South America: “Whatever the truth, Bass was never heard of again.” (ch. 12, “Flinders – Coasting Australia”, pg. 176) That was in 1803. I hadn’t heard of Bass before or of his even more adventurous companion Matthew Flinders. And I didn’t know that Vancouver in Canada was named after the explorer George Vancouver. I’m glad to have changed that.

I had heard of William Bligh, captain of the Bounty, but I’ understood the scale of his achievements better by reading this book. He had witnessed Cook’s death on Hawaii, which was why he didn’t want to risk landing on any of the islands of the Tongan archipelago after he was set adrift in an open boat by Fletcher Christian and his fellow mutineers. Instead, equipped with only a sextant and compass, Bligh set sail for “Timor, in the Dutch East Indies, some 3,600 nautical miles away” (ch. 4, “Bligh’s Boat Journey”, pg. 41). He needed both skill and “bloody-minded determination” to succeed.

He also needed intelligence. That combination explains why this book about mapping the world’s oceans is dominated by men from a small corner of that world: north-western Europe. Cook, Bligh, Flinders and Bass were English; Louis-Antoine de Bougainville and Jean-François de Galaup, comte de Perouse, were French. There’s an “x” in sextant and an “XY” in the human beings who invented and used the instrument. Galileo was one of them: his discovery of the Jovian moons provided a way to determine longitude.

Latitude was relatively easy: you can obtain that by determining the height of, say, Polaris at the north celestial pole. If Polaris is directly overhead, you’re at the north pole. If it’s on the horizon, you’re on the equator. If you can’t see Polaris at all, you’re in the southern hemisphere. Or it’s daylight or a cloudy night. Navigation in past centuries was difficult and dangerous. When Admiral Sir Cloudesley Shovell got it wrong “on the night of 22 October 1707”, he lost four ships and 2,000 men on the “reef-strewn Isles of Scilly” (ch. 5, “Anson’s Ordeals”, pg. 54). Barrie adds that “Shovell himself was washed ashore and reportedly murdered by a local woman who fancied the ring on his finger.”

Even today, with GPS, radar and secure communications, the sea is still claiming lives. This book reminds you of the days when it claimed many more and was a much more frightening place to venture. Those days may return: modern electronics and satellite technology are a fragile system and Barrie describes at the end of the book how some sailors deliberately abandon it, training themselves to rely on their own eyes and brains, not on the pressing of buttons. This book is about balls in more senses than one. The Polynesians who made astonishing voyages over the Pacific didn’t use only their eyes:

When the horizon was obscured and its changing slant could not tell them how their boat was responding to the waves, they apparently stood with their legs apart, using the inertia of their testicles as a guide. (ch. 17, “‘These are men’”, pg. 283)

That’s a reminder of the male biochemistry underlying the courage required to face the sea and the spatial skills that had to accompany it. There are lots of balls elsewhere: the terrestrial globe and the globes of the sun, moon, planets and stars that helped men navigate their way around it. Sextant is a fascinating read about some formidable men and their often frightening voyages. They helped shape the modern world and you can’t understand the modern world without knowing something about them. This book is an excellent place to start.

Infinitesimal is an entertaining read on a fascinating topic: the pioneers of a new form of mathematics and those who opposed them. Amir Alexander claims that “the ultimate victory of the infinitely small helped open the way to a new and dynamic science, to religious toleration, and to political freedoms unknown in human history” (Introduction, pg. 14).

It’s an extraordinary claim and I don’t think he manages to provide extraordinary proof for it. In fact, he probably gets cause-and-effect reversed. Is it likelier that new mathematics opened minds, dynamized science and transformed politics or that open minds created new forms of mathematics, science and politics? I’d suggest that support for the new mathematics was a symptom, not a cause, of a new psychology. But Alexander makes a good case for his thesis and there is no doubt that the world was changed by the willingness of mathematicians to use infinitesimals. Calculus was one result, after all. The book begins in Italy and ends in England, because the pioneers lost in Italy:

For nearly two centuries, Italy had been home to perhaps the liveliest mathematical community in Europe. … But when the Jesuits triumphed over the advocates of the infinitely small, this brilliant tradition died a quick death. With Angeli silenced, and Viviani and Ricci keeping their mathematical views to themselves, there was no mathematician left in Italy to carry on the torch. The Jesuits, now in charge, insisted on adhering close to the methods of antiquity, so that the leadership in mathematical innovation now shifted decisively, moving beyond the Alps, to Germany, England, France and Switzerland. (ch. 5, “The Battle of the Mathematicians”, pg. 178)

Why were the Jesuits involved in an esoteric mathematical dispute? You might say that de minimis curat Loyola – Ignatius Loyola (1491-1556), founder of the Jesuits, cared about anything, no matter how small, that might undermine the authority of the Church. In the view of his successors, the doctrine of indivisibles did precisely that: “in its simplest form, the doctrine states that every line is composed of a string of points, or ‘indivisibles’, which are the line’s building blocks, and which cannot themselves be divided” (Introduction, pg. 9).

Indivisibles must be infinitesimally small, or they wouldn’t be indivisible, but then how does an infinitesimal point differ from nothing at all? And if it isn’t nothing, why can’t it be divided? These paradoxes were familiar to the ancient Greeks, which is why they rejected infinitesimals and laid the foundations of mathematics on what seemed to them to be solider ground. In the fourth century before Christ, Euclid used axioms and rigorous logic to create a mathematical temple for the ages. He proved things about infinity, like the inexhaustibility of the primes, but he didn’t use infinitesimals. When Archimedes broke with Greek tradition and used infinitesimals to make new discoveries, “he went back and proved every one of them by conventional geometrical means, avoiding any use of the infinitely small” (Introduction, pg. 11).

So even Archimedes regarded them as dubious. Aristotle rejected them altogether and Aristotle became the most important pre-Christian influence on Thomas Aquinas and Catholic philosophy. Accordingly, when mathematicians began to look at infinitesimals again, the strictest Catholics opposed the new development. Revolutionaries like Galileo were opposed by reactionaries like Urban VIII.

But the story is complicated: Urban had been friendly to Galileo until “the publication of Galileo’s Dialogue on the Copernican system and some unfavourable political developments” (pg. 301). So I don’t think the mathematics was driving events in the way that Alexander suggests. Copernicus didn’t use them and the implications of his heliocentrism were much more obvious to many more people than the implications of infinitesimals could ever have been. That’s why Copernicus was frightened of publishing his ideas and why Galileo faced the Inquisition for his astronomy, not his mathematics.

But Amir’s thesis makes an even more interesting story: the tiniest possible things had the largest possible consequences, creating a new world of science, politics and art. In Italy, two of the chief antagonists were Galileo and Urban; in England, two were the mathematician John Wallis (1616-1703) and the philosopher Thomas Hobbes (1588-1679). Alexander discusses Wallis and Hobbes in Part II of the book, “Leviathan and the Infinitesimal”. Hobbes thought that de minimis curat rex – “the king cares about tiny things”. Unless authority was absolute and the foundations of knowledge certain, life would be “nasty, brutish and short”.

However, there was a big problem with his reasoning: he thought he’d achieved certainty when he hadn’t. Hobbes repeatedly claimed to have solved the ancient problem of the “quadrature of the circle” – that is, creating a square equal in size to a given circle using only a compass and an unmarked ruler. Wallis demolished his claims, made Hobbes look foolish, and strengthened the case for religious toleration and political freedom. But I don’t think this new liberalism depended on new mathematics. Instead, both were products of a new psychology. Genetics will shed more light on the Jesuits and their opponents than polemics and geometry textbooks from the period. Alexander’s theory is fun but flawed.

In Borges’ story “The Book of Sand” (1975), the narrator acquires a heavy little book that has an infinite number of pages. When he opens it, he can never find the same page twice. The discrepancy between its finite size and its infinite contents begins to prey on his mind. He decides the book is a monstrous thing and wants to get rid of it: “I considered fire, but I feared that the burning of an infinite book might be similarly infinite, and suffocate the planet in smoke.”

It’s a good story, but the central idea doesn’t work, unless you assume magic is at work. A book with an infinite number of pages would be infinitely heavy. In fact, it would instantly become a black hole and start swallowing the universe.

So I assume, anyway. I’m interested in physics but I don’t know much about it. This book is aimed at people like me. It reminded me of Borges’ Book of Sand, partly because it’s small but heavy, partly because of the density of its ideas and the weight of history behind those ideas. Each page of explanation could easily become a hundred or a thousand: physics is daunting in its scope and complexity. Some of the greatest minds in history have put centuries of effort into understanding the behaviour of matter and energy.

That’s how we got astonishing things like electronics, X-rays and the atom bomb. Physics is an intellectual over-achiever, the super-star of the sciences, the most spectacular, powerful and difficult of all. But it’s the most difficult science because it’s also the simplest. Stars and steam-engines are much less complex than societies or brains, which is why you can’t get away with talking nonsense in physics. And although mathematics governs everything, it’s the simpler things – pendulums, light-rays, atoms, stars – that we can mathematize first.

Or some of us can, at least: the highly intelligent and obsessive men, like Galileo and Isaac Newton, who began modern physics by finding ways to extract abstract mathematics from concrete realities. If they’d tried to find maths in psychology or culture, they would have failed, because those things are too complex. They had to look at much simpler things like falling objects, planetary motion and light-rays. Galileo and Newton laid the foundations and later physicists have built on them, so that physics now towers into the scientific skies, the envy and awe of those working with more complex and intractable aspects of existence.

Giles Sparrow takes his readers on a tour of the tower. I suppose you could say he’s operating an express elevator, stopping briefly on the floors and offering a brief explanation of what it contains: elastic and inelastic collisions on one floor, fluid mechanics on another, mass spectrometry, electromagnetic induction and quantum electrodynamics on more. Then the doors snap shut and the elevator shoots up another floor. But one thing is found everywhere: mathematics. Sparrow quotes a lot of equations and uses a lot of numbers. If you want to understand physics, you have to know the maths. If you don’t, there’s no way to disguise your ignorance.

The maths is beyond me, so until brain-modification arrives I won’t be able to understand physics properly. Until then, this book is a good way of glimpsing the glories of the science. It’s also the closest you’ll get to handling Borges’ Book of Sand in real life.

Discovering the Universe: The Story of Astronomy, Paul Murdin (Andre Deutsch 2014)

First published in 2011 as Mapping the Universe, this is a well-written, well-illustrated history of astronomy that begins in the Stone Age and ends with the Hubble Space Telescope and Large Hadron Collider. The photographs will stimulate your eyes as the text stimulates your mind. The universe is a big place and big things happen there, like gamma-ray bursts (GRBs):

Until 1997, astronomers didn’t know whether GRBs originated in some sort of explosions on the edge of our solar system, around our Galaxy, or far away. Two examples proved that the explosions occur the edge of the observable Universe. For their duration of a few seconds, the bursts had been over a million times brighter than their parent galaxy, the biggest bangs since the Big Bang. (ch. 17, “Exploding Stars”, pg. 87)

Ptolemy, Galileo and Newton would all be astonished by the technology that allows modern astronomers to study phenomena like gamma-ray bursts, but one thing has remained constant: the importance of mathematics and measurement in studying the sky. The story of astronomy is not just about seeing further and clearer, but also of measuring better and mathematizing more powerfully. Ptolemy’s geocentric universe entailed the arbitrary complexity of epicycles on epicycles, to explain how the planets sometimes seemed to move backwards against the stars. Then Copernicus resurrected the ancient Greek hypothesis of a heliocentric universe.

Back cover

Planetary retrogression became easier to explain. Other hypotheses, like the steady state universe and Kepler’s planetary Platonic solids, haven’t proved successful, but data don’t explain themselves and astronomers have to be adventurous in mind, if not usually in body. This book contains the big names, the big sights and the big mysteries that are still awaiting explanation. More big names, sights and mysteries are on their way.

Caricatures are compelling because they simplify and exaggerate. A good artist can create one in a few strokes. In fact, a good artist has to caricature if he can use only a few strokes. The image won’t be recognizable otherwise.

This also applies to philosophical ideas. If you have to describe them in relatively few words, you’ll inevitably caricature, making them distinct but losing detail and complexity. So this book is a series of caricatures. With only 382 pages of standard print, what else could it be? In each case, Philip Stokes uses a few strokes to portray “100 Essential Thinkers” from Thales of Miletus, born c. 620 B.C., to William Quine (1908-2000), with all the big names in between: Plato, Aristotle, Descartes, Pascal, Hume, Kant, Leibniz, Schopenhauer, Nietzsche, Russell, Wittgenstein and so on. The philosophical portraits are recognizable but not detailed. But that’s why they’re fun, like a caricature.

It’s also fun to move so quickly through time. There are nearly three millennia of Western philosophy here, but the schools and the civilizations stream by, from the Pre-Socratics and Atomists to the Scholastics and Rationalists; from pagan Greece and Rome to Christianity and communism. Bertrand Russell’s History of Western Philosophy, which inevitably comes to mind when you look at an over-view like this, moves much more slowly, but it’s a longer and more detailed book.

It’s also funnier and less inclusive. This book discusses men who are more usually seen as scientists or mathematicians, like Galileo and Gödel. But in a sense any historic figure could be included in an over-view of philosophy, because everyone has one. You can’t escape it. Rejecting philosophy is a philosophy too. Science and mathematics have philosophical foundations, but in some ways they’re much easier subjects. They’re much more straightforward, like scratching your right elbow with your left hand.

Philosophy can seem like trying to scratch your right elbow with your right hand. The fundamentals of existence are difficult to describe, let alone understand, and investigating language using language can tie the mind in knots. That’s why there’s a lot of room for charlatans and nonsense in philosophy. It’s easier to pretend profundity than to be profound. It’s also easy to mistake profundity for pseudery.

And, unlike great scientists or mathematicians, great philosophers should be read in the original. Reading Nietzsche in English is like looking at a sun-blasted jungle through tinted glass or listening to Wagner wearing earplugs. Or so I imagine: I can’t read him in German. But some philosophers suffer less by translation than others, because some philosophical ideas are universal. Logic, for example. But how important is logic? Is it really universal? And is mathematics just logic or is it something more?

You can ask, but you may get more answers than you can handle. Philosophy is a fascinating, infuriating subject that gets everywhere and questions everything. You can’t escape it and this book is a good place to learn why.

Mapping the World: The Story of Cartography, Beau Riffenburgh (Carlton Books 2011, 2014)

A good map is like a swan on a river. Beneath the elegance there is a lot of effort. This book is about that effort: all the millennia of research and refinement that have gone into perfecting maps. Not that any map can be perfect. As Beau Riffenburgh explains here, there are always choices to be made: what do you put in, what do you leave out? And how do you represent spherical geometry on flat paper?

The Flemish cartographer Gerardus Mercator came up with one famous answer to that question:

Mercartor’s major achievement came in 1569 with a new projection that represented a breakthrough in nautical cartography. Since known as the Mercator projection, it is cylindrical-like, with the meridians as equally spaced parallel lines and the lines of latitude as parallel, horizontal lines, which are spaced further apart as their distance from the equator increases. This projection is uniquely suited to navigation because a line of constant true bearing allows a navigator to plot a straight-line course. However, this projection grossly distorts geographical regions in high latitudes – thus Greenland is shown larger than South America, although it is actually less than one-eighth of the size. (“Cosmographies and the Development of Projection”, pg. 51)

So the map looks wrong, but leads right. So does the famous map of the London Underground, which ignores true distances and bearings: the designer Harry Beck made it look like an “electrical circuit, with straight lines and the inclusion of only one feature above ground – the Thames” (“Mapping for the Masses”, pg. 143). Maps are about abstraction: they condense and confine what people find interesting or important about the real world.

So minds mould maps and in writing about maps, Riffenburgh is also writing about culture and politics. About art too, because maps can be very beautiful things, sometimes deliberately, sometimes incidentally. Above all, however, he’s writing about mathematics. What was implicit from the beginning – the importance of maths in mapping – became more and more explicit, as he describes in the chapter “Men, Measurements and Mechanisms” (pp. 70-3). The men are drawn from the world’s most evil and energetic group: white Europeans. Galileo, Newton and Huygens are three of them: as they contributed to maths and science, they contributed to cartography.

Another man is the Yorkshire watchmaker John Harrison (1693-1776), the hero of Dava Sobel’s Longitude (1995). He was a remarkable personality and looks it in the portrait here: proud, determined and self-possessed. He needed all those qualities to get his due. He invented a chronometer that kept accurate time on long voyages and enabled navigators to determine longitude, but British officialdom “made him wait years for all of his prize-money” (pg. 73).

Elsewhere the names are obscurer and the stories sometimes sadder:

In the history of cartography, few individuals stand out for their work in so many geographical regions and aspects of science as James Rennell. Born in Devon in 1742, Rennell went to sea at the age of 14, learned maritime surveying and then, at the end of the Seven Years’ War, received a commission in the Bengal Army as an engineer. … Equipped with quadrant, compass and chain, Rennell began a thorough and scientific survey of [Bengal’s] major river systems, roads, plains, jungles, mangrove forests and mountains. (“James Rennell: Mapping India, Africa and Ocean Currents”, pg. 86)

However, he “never fully recovered from a severe wound received in an ambush” and retired to London to produce his “masterpiece – A Map of Hindoostan, or the Mogul Empire” (1782/1788). But en route to England, he had an “extended stay in Southern Africa” and developed an interest in ocean currents. So he became a pioneering hydrographer too: his posthumous An Investigation of Currents of the Atlantic Ocean (1832) “is often considered to form the historical basis of the study of currents” (pg. 89).

Later in the century, the German August Petermann worked for the Royal Geographical Society and was appointed “Physical Geographer Royal” by Queen Victoria. His assistant John Bartholomew said “no one has done more than he to advance modern cartography”, but Petermann committed suicide in 1878 after returning to Germany (“Maps reach a wider audience”, pg. 132).

Nietzsche would not have approved. But I think he would have applauded this:

Perhaps the most remarkable nautical drawings of all, considering the conditions under which they were produced, were those of William Bligh, captain of the British ship HMAV [His Majesty’s Armed Vessel] Bounty in 1789. Following the infamous mutiny, Bligh and 18 loyal seamen were set adrift in the ship’s launch. During the next 47 days, Bligh navigated approximately 3,600 nautical miles (6,660 km) to Timor, with only one stop. Throughout the journey, which is considered one of the most remarkable accomplishments in the history of open-boat travel, Bligh kept a detailed log and made sketches of his course. (“Mapping Australia and the Pacific”, pg. 77)

His chart is reproduced here. Using anecdotes like that with serious analysis and intellectual history, Riffenburgh tells the story of cartography from Mesopotamia and before to the moon and beyond. The story of maps is the story of man: even pre-literate societies like the ancient Polynesians have used maps to record the sea and its currents. In Europe, maps have reflected every advance in technology, like printing and photography. But as they’ve responded to technology, they’ve altered the way we see and interact with reality. When you look at a map, there’s a whole world of exploration, endeavour and ingenuity just beyond its margins. Mapping the World is about that world: the margins of mapness without which the maps themselves would not exist. It’s a book to stimulate the mind and delight the eye.